Characterization of g∞,σ‐integral operators

Arango, G.; Molina, Juan Antonio López; Rivera, M. J.

doi:10.1002/mana.200310286

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…e. µ j γ = µ n+1 γ in M γ , 1 ≤ j ≤ n + 1. Then "piecing together" the measure spaces (Ω γ , M γ , µ n+1 γ ) and following a standard procedure (see [1] and proposition 4.5, step 1 in [2] for more details) we obtain a localizable measure space (Ω, M, µ n+1 ) and surjective isometric order isomorphisms…”

Section: Claim 1 Each V J Is a Maximal System Of Pairwise Disjoint Wmentioning

confidence: 99%

Multilinear Operator Ideals Associated to Saphar Type (n + 1)-tensor Norms

Molina¹

2006

Positivity

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We study an (n + 1)-tensor norm gr,s generalizing Saphar's classic norm to (n+1)-fold tensor products. We characterize the maps in the minimal and maximal multilinear operator ideals related to gr,s in the sense of Defant and Floret.Mathematics Subject Classification. Primary 46M05; 46A32. Keywords. (n + 1)-fold tensor products, gr,s-nuclear and gr,s-integral n-linear operators, t-absolutely summing n-linear operators, Ultraproducts.In [11] Pietsch proposed building a systematic theory of ideals of multilinear mappings between Banach spaces, similar to the already well-developed one regarding linear maps, as a first step to study ideals of more general non linear operators.A systematic development of this program has not been initiated until the works [4] and [5] of Floret, mainly motivated by the potential applications of the new theory to infinite holomorphy. Thus, classic notions of maximal operator ideals with their associated α-tensor norm, dual tensor norm α and the related α-nuclear and α-integral operators can be extended to the framework of multilinear operator ideals and multiple tensor products.However, as far as we know, only a few concrete examples of multi-tensor norms have been considered to check and apply the general concepts of the extended theory beyond the injective and projective ones or the introduction of generalized α-nuclear operators (for example, the previous work in [9] stops at this point). In this sense we have studied in [8] an (n + 1)-fold tensor norm generalizing both Lapresté's classic norm as well as Matos's multi-tensor norm defined in [9].The purpose of this paper is to study an (n + 1)-fold tensor norm naturally extending the well-known tensor norm g p of Saphar. This new tensor norm on tensor products n+1 j=1 E j , 1 ≤ n ∈ N, of n + 1 normed spaces E j , will depend on two vector parameters r := (r 1 , r 2 , · · · , r k ) and s := (r k+1 , r k+2 , · · · , r n+1 ) and will be Partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.

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Section: Claim 1 Each V J Is a Maximal System Of Pairwise Disjoint Wmentioning

confidence: 99%

Multilinear Operator Ideals Associated to Saphar Type (n + 1)-tensor Norms

Molina¹

2006

Positivity

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“…This result shows the role of the real interpolation method in the description of these operators. Following this line of research, the papers of López Molina and Sánchez Pérez [12,13], Arango, López Molina, and Rivera [1] completed the study of this operator ideal using the LionsPeetre interpolation theory and the so-called trace duality (see also [20]). The analysis of the relation between this operator ideal and cotype on Banach spaces has been also developed in [21].…”

Section: Example 33mentioning

confidence: 99%

“…The Maurey-Rosenthal Theorem can also be understood as a result involving strong convergence of sequences, i.e., for a σ-order continuous q-convex Köthe function space X(µ) and a q-concave operator T , there is a µ-continuous finite measure µ 0 such that (1) …”

Section: Introductionmentioning

confidence: 99%

Asymptotic domination of operators on Köthe function spaces and convergence of sequences

Pérez

2006

Mathematische Nachrichten

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We study the asymptotic behavior of Maurey-Rosenthal type dominations for operators on Köthe function spaces which satisfy norm inequalities that define weak q-concavity properties. In particular, we define and study two new classes of operators that we call α-almost q-concave and qα-concave operators (1 ≤ q < ∞, 0 ≤ α < 1). We also provide a factorization theorem through real interpolation spaces for qα-concave operators. We also discuss some direct consequences of these results regarding the strong convergence of sequences on Köthe function spaces.

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“…This relation was used in [16] for the case 1 < p < ∞, where the ideals of nuclear and integral operators corresponding to a tensor norm were characterized by means of factorizations through interpolation spaces. The case p = 1 was studied in [2]. These works put in evidence the role interpolation spaces may play in defining tensor norms that are more general than the classical ones.…”

Section: Introductionmentioning

confidence: 99%

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces

Puerta

Loaiza

2010

Can. math. bull.

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Abstract. The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

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Characterization of g_∞,σ‐integral operators

Cited by 4 publications

References 11 publications

Multilinear Operator Ideals Associated to Saphar Type (n + 1)-tensor Norms

Multilinear Operator Ideals Associated to Saphar Type (n + 1)-tensor Norms

Asymptotic domination of operators on Köthe function spaces and convergence of sequences

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces

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